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A short exact sequence of abelian coefficient sheaves induces a long exact sequence of abelian Čech cohomologies with coefficients in those, provided we have some topological assumption satisfied. The failure in general is due to the fact that the global sections for sheaves is only a left exact and not an exact functor, so one does not have the short exact sequence at the level of groups for fixed cover, and hence no long exact sequence; this can be repaired if one gets to sufficiently fine local cover where global sections are well behaved. For example things work for the algebraic varieties in Zariski topology, as well as for all paracompact Hausdorff spaces. In full generality, it is however not true, and the long exact sequence has to be modified. There is a careful treatment (paragraph 24 for the general case) in Serre’s FAC article where he introduces something he calls $H_0$ instead of $H$ to repair the sequence.
Is there something more modern about the issue ? Is this related to the issues of hypercover versus Čech discussion elsewhere or this is a different issue at hand ? Do we have any related discussion in $n$Lab ?
A short exact sequence of abelian coefficient sheaves induces a long exact sequence of abelian Čech cohomologies with coefficients in those, provided we have some topological assumption satisfied. The failure in general is due to the fact that the global sections for sheaves is only a left exact and not an exact functor, so one does not have the short exact sequence at the level of groups for fixed cover, and hence no long exact sequence;
This is not quite correct: it is true that forming Cech cocycles will not in general send short exact sequences to short exact sequences of cocycles, but when those topological conditions are satisfied, that you allude to, it will always send it directly to a long fiber sequence.
Notice that a short exact sequence of cocycles is not homotopy-meaningful anyway. Cocycles form an $\infty$-groupoid and it makes sense to say they sit in a fiber sequence, not in a short exact sequence.
More in detail: if $A \to B \to C$ is a short exact sequence of group objects in $Sh(C)$, then $\mathbf{B}^q A \to \mathbf{B}^q B \to \mathbf{B}^q C$ is a homotopy fiber sequence in $\mathbf{H} := Sh_\infty(C)$ for all $n$. Therefore for all $X \in \mathbf{H}$ also $\mathbf{H}(X, \mathbf{B}^q A) \to \mathbf{H}(X, \mathbf{B}^q B)\to \mathbf{H}(X, \mathbf{B}^q C)$ is a fiber sequence (now in $\infty Grpd$) and so we get a long sequence in cohomology
$\cdots \to H^{q-1}(X, C) \to H^q(X, A) \to H^q(X, B) \to H^q(X, C) \to H^{q+1}(X, A) \to \cdots \,.$Cech cocycles are a way to model $\mathbf{H}(X,-)$, with those “topological assumpotions” being sufficient conditions for making this work. So if they are satisfied, we always have the long exact sequence in cohomology induced from a short exact sequence of coefficient groups.
It will not, in general. come from a short exact sequence of cocycles. But that is something evil to ask for anyway.
Well, cocycle level was here just a hint. Of course, thank you for pointing out the true meaning as fiber sequence for the case when things work.
I am clearly interested in what is known (since Serre) when the topological assumptions are not satisfied. The solution in Serre, with introducing “$H_0$”, looks a bit like a hack to me, and I would expect something more by nowdays. So my question is still posed.
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